The Theory of Rings

Rings are worn around the finger as ornamental jewelry. They can be made of almost any hard material and may be set with gems or other materials. They fit snugly around the finger, so bands that are worn loosely like bracelets are not considered rings. In some cultures, rings are also used as symbols of marriage, exceptional achievement, high status or authority, membership in an organization, and the like.

The earliest rings were simple, usually round, metal bands with no decoration. Over time, they became more elaborate and were often decorated with carvings or engravings. In modern times, rings can be fashioned from almost any material and are commonly made of precious metals such as gold and silver. Some of them are set with gemstones such as diamonds, sapphires and rubies, or may be made to look like natural stones such as jade. Some are also made of pearls or synthetic materials such as cubic zirconia.

For many people, the most important reason for wearing a ring is to show that they are married or engaged. In addition, some people wear rings as a symbol of their faith or cultural heritage, and others believe that certain types of rings bring good luck or protection from evil.

Most people choose to wear a wedding ring on the left hand because it has traditionally been a symbol of marriage. However, some women wear a wedding ring on the right hand, and in some cultures it is customary for men to wear a ring on both hands. In some cases, this is to signify that they are married or in a relationship, while in other cases it may be a way to show their commitment to each other.

While most rings are circular in shape, they can also be square, oval or rectangular. Some have a double-halo, meaning that there are two layers of smaller diamonds surrounding the center stone. This design can make a small diamond appear larger, and is a popular choice for engagement rings. Other rings have a bezel setting, which holds the stone securely in place and makes it nearly impossible for the ring to get lost or stolen.

The theory of rings is an important area of algebra, and it has connections to other branches of mathematics, especially number theory and algebraic geometry. The concept of a ring is central to the field of combinatorics, and the notion of a ringed topos is an important generalization of the standard set-theoretic definition of a ring.

The rings of the cyclic groups Z nmathbbZn under their mod-nn operations are examples of basic rings, and they are generalized to monoids internal to abelian groups and more generally stable infinity-categories. Rings are also of interest in algebraic topology, where they play a key role in the construction of homology and cohomology theories. In fact, Noether’s work on this subject was influenced by her ideas about rings. Until recently, most (but not all) books on algebra followed Noether’s convention and did not require a 1 in the definition of a ring. However, beginning with the 1960s, it has become common for advanced algebra texts to define rings as monoids.